We present a generic, semi-automated algorithm for generating non-uniform coarse grids for modeling subsurface flow. The method is applicable to arbitrary grids and does not impose smoothness constraints on the coarse grid. One therefore avoids conventional smoothing procedures that are commonly used to ensure that the grids obtained with standard coarsening procedures are not too rough. The coarsening algorithm is very simple and essentially involves only two parameters that specify the level of coarsening. Consequently the algorithm allows the user to specify the simulation grid dynamically to fit available computer resources, and, e.g., use the original geomodel as input for flow simulations. This is of great importance since coarse grid-generation is normally the most time-consuming part of an upscaling phase, and therefore the main obstacle that has prevented simulation workflows with userdefined resolution. We apply the coarsening algorithm to a series of two-phase flow problems on both structured (Cartesian) and unstructured grids. The numerical results demonstrate that one consistently obtains significantly more accurate results using the proposed non-uniform coarsening strategy than with corresponding uniform coarse grids with roughly the same number of cells.

Vugs, caves, and fractures can significantly alter the effective permeability of carbonate reservoirs and should be accurately accounted for in a geomodel. Accurate modeling of the interaction between free-flow and porous regions is essential for flow simulations and detailed production engineering calculations. However, flow simulation of such reservoirs is very challenging because of the co-existence of porous and free-flow regions on multiple scales that need to be coupled.Multiscale methods are conceptually well-suited for this type of modeling as they allow varying resolution and provide a systematic procedure for coarsening and refinement. However, to date there are hardly no multiscale methods developed for problems with both free-flow and porous regions. Our work is a first step to make a uniform multiscale framework, where we develop a multiscale mixed finite-element (MsMFE) method for detailed modeling of vuggy and naturally-fractured reservoirs. The MsMFE method uses a standard Darcy model to approximate pressure and fluxes on a coarse grid, but captures fine-scale effects through basis functions determined from numerical solutions of local Stokes-Brinkman flow problems on the underlying fine-scale geocellular grid. The Stokes-Brinkman equations give a unified approach to simulating free-flow and porous regions using a single system of equations, avoid explicit interface modeling, and reduce to Darcy or Stokes flow by appropriate choices of parameters.In the paper, the MsMFE solutions are compared with fine-scale Stokes-Brinkman solutions for test cases including both short-and long-range fractures. The results demonstrate how fine-scale flow in fracture networks can be represented within a coarse-scale Darcy flow model by using multiscale elements computed solving the Stokes-Brinkman equations. The results indicate that the MsMFE method is a promising path toward direct simulation of highly detailed geocellular models of vuggy and naturally-fractured reservoirs. IntroductionNaturally fractured and carbonate reservoirs are composed of porous material, but will typically also contain relatively large void spaces in the form of fractures, small cavities, and caves, which are called vugs in the geological literature. Flow simulation of such formations is very challenging because of the co-existence of porous and free-flow domains on multiple scales that require coupling (Wu et al. 2006).The Darcy-Stokes equations have been used to model industrial infiltration processes and coupled surface and subsurface flow, for which the porous and the free-flow domains are well separated. The Darcy-Stokes model consists of Darcy's law combined with mass conservation in the porous subdomain and the Stokes equations in the free-flow subdomain. To close the model, one must specify conditions on the interface between the Darcy and Stokes subdomains. All these conditions require continuity of mass and momentum over the interface, but differ in the way they allow the tangential component to jump across the interface.In a ca...

Geological models are becoming increasingly large and detailed to account for heterogeneous structures on different spatial scales. To obtain simulation models that are computationally tractable, it is common to remove spatial detail from the geological description by upscaling. Pressure and transport equations are different in nature and generally require different strategies for optimal upgridding. To optimize the accuracy of a transport calculation, the coarsened grid should generally be constructed based on a posteriori error estimates and adapt to the flow patterns predicted by the pressure equation. However, sharp and rigorous estimates are generally hard to obtain, and herein we therefore consider various ad hoc methods for generating flow-adapted grids. Common for all is that they start by solving a single-phase flow problem once and then continue to form a coarsened grid by amalgamating cells from an underlying fine-scale grid. We present several variations of the original method. First, we discuss how to include a priori information in the coarsening process, e.g. to adapt to special geological features or to obtain less irregular grids in regions where flowadaption is not crucial. Second, we discuss the use of bi-directional versus net fluxes over the coarse blocks and show how the latter gives systems that better represent the causality in the flow equations, which can be exploited to develop very efficient nonlinear solvers. Finally, we demonstrate how to improve simulation accuracy by dynamically adding local resolution near strong saturation fronts.

Summary We develop an adjoint model for a simulator consisting of a multiscale pressure solver and a saturation solver that works on flow-adapted grids. The multiscale method solves the pressure on a coarse grid that is close to uniform in index space and incorporates fine-grid effects through numerically computed basis functions. The transport solver works on a coarse grid adapted by a fine-grid velocity field obtained by the multiscale solver. Both the multiscale solver for pressure and the flow-based coarsening approach for transport have shown earlier the ability to produce accurate results for a high degree of coarsening. We present results for a complex realistic model to demonstrate that control settings based on optimization of our multiscale flow-based model closely match or even outperform those found by using a fine-grid model. For additional speed-up, we develop mappings used for rapid system updates during the timestepping procedure. As a result, no fine-grid quantities are required during simulations and all fine-grid computations (multiscale basis functions, generation of coarse transport grid, and coarse mappings) become a preprocessing step. The combined methodology enables optimization of waterflooding on a complex model with 45,000 grid cells in a few minutes.

Vugs, caves, and fractures can significantly alter the effective permeability of carbonate reservoirs and should be accurately accounted for in a geomodel. Accurate modeling of the interaction between free-flow and porous regions is essential for flow simulations and detailed production engineering calculations. However, flow simulation of such reservoirs is very challenging because of the coexistence of porous and free-flow regions on multiple scales that need to be coupled. Multiscale methods are conceptually well-suited for this type of modeling as they allow varying resolution and provide a systematic procedure for coarsening and refinement. However, to date there are hardly no multiscale methods developed for problems with both free-flow and porous regions. Herein we develop a multiscale mixed finite-element (MsMFE) method for detailed modeling of vuggy and naturally-fractured reservoirs as a first step towards a uniform multiscale, multiphysics framework. The MsMFE method uses a standard Darcy model to approximate pressure and fluxes on a coarse grid, whereas fine-scale effects are captured through basis functions computed numerically by solving local Stokes-Brinkman flow problems on the underlying fine-scale geocellular grid. The Stokes-Brinkman equations give a unified approach to simulating free-flow and porous regions using a single system of equations, avoid explicit interface modeling, and reduce to Darcy or Stokes flow in certain parameter limits. In the paper, the MsMFE solutions are compared with fine-scale Stokes-Brinkman solutions for test cases including both short-and long-range fractures. The results demonstrate how fine-scale flow in fracture networks can be represented within a coarse-scale Darcy flow model by using multiscale elements computed solving the Stokes-Brinkman equations. The results indicate that the MsMFE method is a promising path toward direct simulation of highly detailed geocellular models of vuggy and naturally-fractured reservoirs.